Characterizing translation-invariant Bell inequalities using tropical algebra and graph polytopes

Physical Review A, vol. 113

Abstract

Nonlocality is one of the key features of quantum physics, which is revealed through the violation of a Bell inequality. In large multipartite systems, nonlocality characterization quickly becomes a challenging task. A common practice is to make use of symmetries, low-order correlators, or exploiting local geometries, to restrict the class of inequalities. In this paper, we characterize translation-invariant (TI) Bell inequalities with finite-range correlators in one-dimensional geometries. We introduce a novel methodology based on tropical algebra tensor networks and highlight its connection to graph theory. Surprisingly, we find that the TI Bell polytope has a number of extremal points that can be uniformly upper-bounded with respect to the system size. We give an efficient method to list all vertices of the polytope for a particular system size, and characterize the tightness of a given TI Bell inequality. The connections highlighted in our work allow us to re-interpret concepts developed in the fields of tropical algebra and graph theory in the context of Bell nonlocality, and vice-versa. This work extends a parallel article [M. Hu \textit{et al.}, arXiv: 2208.02798 (2022)] on the same subject.

Authors 3

1. Eloïc Vallée Aachen

   Leiden University

   Affiliation as printed

   Applied Quantum Algorithms Leiden

   Universiteit Leiden

2. Tim Seynnaeve

   KU Leuven

   Affiliation as printed

   KU Leuven

3. Patrick Emonts Aachen

   Leiden University

   Affiliation as printed

   Applied Quantum Algorithms Leiden

   Universiteit Leiden